It is a basic theorem in modern quantum information theory.
Ruskai,[3] building on results obtained by Lieb in his proof of the Wigner-Yanase-Dyson conjecture.
The proof of this relation in the classical case is quite easy, but the quantum case is difficult because of the non-commutativity of the reduced density matrices describing the quantum subsystems.
Some useful references here include: We use the following notation throughout the following: A Hilbert space is denoted by
Density matrices on a tensor product are denoted by superscripts, e.g.,
is Umegaki's[8] quantum relative entropy of two density matrices
It states that This inequality is true, of course, in classical probability theory, but the latter also contains the theorem that the conditional entropies
is the Araki–Lieb triangle inequality [9] which is derived in [9] from subadditivity by a mathematical technique known as purification.
Equivalently, the statement can be recast in terms of conditional entropies to show that for tripartite state
The strong subadditivity inequality was improved in the following way by Carlen and Lieb [10] with the optimal constant
[4] It was this paper that provided the mathematical basis of the proof of SSA by Lieb and Ruskai.
[13] The theorem can also be obtained by proving numerous equivalent statements, some of which are summarized below.
E. P. Wigner and M. M. Yanase [14] proposed a different definition of entropy, which was generalized by Freeman Dyson.
in [4] in the following more general form: The function of two matrix variables is jointly concave in
The relative entropy decreases monotonically under completely positive trace preserving (CPTP) operations
The most important and basic class of CPTP maps is a partial trace operation
Then which is called Monotonicity of quantum relative entropy under partial trace.
Since the trace (and hence the relative entropy) is unitarily invariant, inequality (3) now follows from (2).
Since the extreme points of the convex set of density matrices are pure states, SSA follows from JC; See,[21][22] for a discussion.
In,[23][24] D. Petz showed that the only case of equality in the monotonicity relation is to have a proper "recovery" channel: For all states
D. Petz also gave another condition [23] when the equality holds in Monotonicity of quantum relative entropy: the first statement below.
Winter described the states for which the equality holds in SSA.
satisfies strong subadditivity with equality if and only if there is a decomposition of second system as into a direct sum of tensor products, such that with states
Carlen have found an explicit error term in the SSA inequality,[10] namely,
Then, in this "highly quantum" regime, this inequality provides additional information.
In his paper [26] I. Kim studied an operator extension of strong subadditivity, proving the following inequality: For a tri-partite state (density matrix)
M. B. Ruskai describes this work in details in [28] and discusses how to prove a large class of new matrix inequalities in the tri-partite and bi-partite cases by taking a partial trace over all but one of the spaces.
A significant strengthening of strong subadditivity was proved in 2014,[29] which was subsequently improved in [30] and.
These improvements of strong subadditivity have physical interpretations in terms of recoverability, meaning that if the conditional mutual information
These results thus generalize the exact equality conditions mentioned above.